# Lognormal distribution

Format: Lognormal(m, s)

The lognormal distribution is one of the most widely used distributions in risk analysis.

Examples of two Lognormal distribution are given below:

## Uses

The Lognormal distribution is useful for modeling naturally occurring variables that are the product of a number of other naturally occurring variables. Central Limit Theorem shows that the product of a large number of independent random variables is Lognormally distributed. For example, the volume of gas in a petroleum reserve is often Lognormally distributed because it is the product of the area of the formation, its thickness, formation pressure, porosity and the gas:liquid ratio.

Lognormal distributions often provide a good representation for a physical quantity that extend from zero to + infinity and is positively skewed, perhaps because some Central limit Theorem type of process is determining the variable's size. Lognormal distributions are also very useful for representing quantities that are thought of in orders of magnitude. For example, if a variable can be estimated to within a factor of 2 or to within an order of magnitude, the Lognormal distribution is often a reasonable model.

Lognormal distributions have also been used to model lengths of words and sentences in a document, particle sizes in aggregates, critical doses in pharmacy and incubation periods of infectious diseases, but one reason the Lognormal distribution appears so frequently is because it is easy to fit  and test (one simply transforms the data to logs and manipulate as a Normal distribution), and so observing its use in your field does not necessarily mean it is a good model: it may just have been a convenient one. Modern software and statistical techniques have removed much of the need for assumptions of normality, so be cautious about using the Lognormal just because it has always been that way.

A variable is Lognormally distributed when the log (to any base) of the variable is Normally distributed i.e:

X is Lognormally distributed if logk[X] is Normally distributed,

Or equivalently X = k^Normal(...) where k is some constant

It is very occasionally known as the Galton-McAlister distribution and, in economics, as the Cobb-Douglas distribution where it is applied to production data.

## Alternative parametrizations

The LognormalE distribution has as its parameters the mean and standard deviation of the Normal distribution of loge[X]..

The LognormalB distribution has as its parameters the mean and standard deviation of the Normal distribution of logB[X]..

The LognormalAlt distribution determines a Lognormal distribution defined by two percentiles.

## ModelRisk functions added to Microsoft Excel for the Lognormal distribution

VoseLogNormal generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseLogNormalObject constructs a distribution object for this distribution.

VoseLogNormalProb returns the probability density or cumulative distribution function for this distribution.

VoseLogNormalProb10 returns the log10 of the probability density or cumulative distribution function.

VoseLogNormalFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseLogNormalFitObject constructs a distribution object of this distribution fitted to data.

VoseLogNormalFitP returns the parameters of this distribution fitted to data.

## Lognormal distribution equations

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##### For Microsoft Excel

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